With commuting Killing vectors, the lapse and shift of one Killing vector are constants along the other
Niall O Murchadha
备注：
The key result in this paper was previously discovered by Beig and Chrusciel using a different approach. I add the appropriate reference and a short discussion of their derivation
Given an d-dimensional manifold with two commuting Killing vectors, together with an d - 1 dimensional submanifold in which one of the Killing vectors lies, then the lapse and shift of the second Killing vector, relative to this slice, remain constant along the orbits of the `surface' Killing vector. Alternatively, the six dot products that can be formed from the three vectors, the two Killing vectors and the normal to the submanifold, are all constants along the `surface' Killing vector.
We study the properties of a modified version of the Bona-Masso family of hyperbolic slicing conditions. This modified slicing condition has two very important features: In the first place, it guarantees that if a spacetime is static or stationary, and one starts the evolution in a coordinate system in which the metric coefficients are already time independent, then they will remain time independent during the subsequent evolution, {\em i.e.} the lapse will not evolve and will therefore not drive the time lines away from the Killing direction. Second, the modified condition is naturally adapte…
We study space-time Killing vectors in terms of their "lapse and shift" relative to some spacelike slice. We give a necessary and sufficient condition in order for these lapse-shift pairs, which we call Killing initial data (KID'S), to form a Lie algebra under the bracket operation induced by the Lie commutator of vector fields on space-time. This result is applied to obtain a theorem on the periodicity of orbits for a class of Killing vector fields in asymptotically flat space-times.
Four-dimensional Riemannian spacetimes with two commuting spacelike Killing vectors are studied in Einstein's theory of gravity, and found that no outer apparent horizons exist, provided that the dominant energy condition holds.
A new first integral for the equations corresponding to twisting type-N vacuum gravitational fields with two non-commuting Killing vectors is introduced. A new reduction of the problem to a complex second-order ordinary differential equation is given. Alternatively, the mentioned first integral can be used in order to provide a first integral of the second-order complex equation introduced in a previous treatment of the problem.
In this paper we derive the most general first-order symmetry operator commuting with the Dirac operator in all dimensions and signatures. Such an operator splits into Clifford even and Clifford odd parts which are given in terms of odd Killing-Yano and even closed conformal Killing-Yano inhomogeneous forms respectively. We study commutators of these symmetry operators and give necessary and sufficient conditions under which they remain of the first-order. In this specific setting we can introduce a Killing-Yano bracket, a bilinear operation acting on odd Killing-Yano and even closed conformal…
We show that puncture data for quasicircular binary black hole orbits allow a special gauge choice that realizes some of the necessary conditions for the existence of an approximate helical Killing vector field. Introducing free parameters for the lapse at the punctures we can satisfy the condition that the Komar and ADM mass agree at spatial infinity. Several other conditions for an approximate Killing vector are then automatically satisfied, and the 3-metric evolves on a timescale smaller than the orbital timescale. The time derivative of the extrinsic curvature however remains significant. …
At Most One Inner Killing Horizon in Stationary Black Holes
Tianjin University（天津大学）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Hangzhou Institute for Advanced Study, UCAS（杭州高等研究院，中国科学院大学）
Recently, the authors have formulated and explored a novel Painleve-Gullstrand variant of the Lense-Thirring spacetime, which has some particularly elegant features -- including unit-lapse, intrinsically flat spatial 3-slices, and some particularly simple geodesics, the "rain" geodesics. At linear level in the rotation parameter this spacetime is indistinguishable from the usual slow-rotation expansion of Kerr. Herein, we shall show that this spacetime possesses a nontrivial Killing tensor, implying separability of the Hamilton-Jacobi equation. Furthermore, we shall show that the Klein-Gordon …
We introduce the domain structure for stationary black hole space-times. Given a set of commuting Killing vector fields of the space-time the domain structure lives on the submanifold where at least one of the Killing vector fields have zero norm. Depending on which Killing vector field has zero norm the submanifold is naturally divided into domains. A domain corresponds either to a set of fixed points of a spatial symmetry or to a Killing horizon, depending on whether the characterizing Killing vector field is space-like or time-like near the domain. The domain structure provides invariants o…
Einstein's equations with two commuting Killing vectors and the associated Lax pair are considered. The equations for the connection $A(ς, η, γ)=Ψ_{,γ}Ψ^{-1}$, where $γ$ the variable spectral parameter are considered. A transition matrix ${\cal T}= A(ς, η, γ)A^{-1}(ξ, η, γ)$ for $A$ is defined relating $A$ at ingoing and outgoing light cones. It is shown that it satisfies equations familiar from integrable pde's theory. A transition matrix on $ς={\mbox constant}$ is defined in an analogous manner. These transition matrices allow us to obtain a hierarchy of integrals of motion with respect to t…
We consider four-dimensional, Riemannian, Ricci-flat metrics for which one or other of the self-dual or anti-self-dual Weyl tensors is type-D. Such metrics always have a valence-2 Killing spinor, and therefore a Hermitian structure and at least one Killing vector. We rederive the results of Przanowski and collaborators, that these metrics can all be given in terms of a solution of the $SU(\infty)$-Toda field equation, and show that, when there is a second Killing vector commuting with the first, the method of Ward can be applied to show that the metrics can also be given in terms of an axisymm…
We consider the Klein-Gordon equation in generalized higher-dimensional Kerr-NUT-(A)dS spacetime without imposing any restrictions on the functional parameters characterizing the metric. We establish commutativity of the second-order operators constructed from the Killing tensors found in arXiv:hep-th/0612029 and show that these operators, along with the first-order operators originating from the Killing vectors, form a complete set of commuting symmetry operators (i.e., integrals of motion) for the Klein-Gordon equation. Moreover, we demonstrate that the separated solutions of the Klein-Gordo…
Starting with the most general four-dimensional spacetime possessing two commuting Killing vectors and a nontrivial Killing tensor, we analytically integrate Einstein-Yang-Mills equations for a completely arbitrary gauge group. It is assumed that the gauge field inherits the symmetries of the background and is aligned with the principal null directions of the spacetime.
Under the action of the Geroch group, the Minkowski metric can be transformed into any vacuum metric with two commuting Killing vectors. In principle, this reduces the problem of deriving vacuum metrics with two commuting Killing vectors to pure algebra. In this short note, we use these facts to give a purely algebraic derivation of the Einstein-Rosen metric, which describes a cylindrical gravitational wave. Our derivation has a straightforward extension to gravitational pulse waves.
We investigate solutions of new massive gravity with two commuting Killing vectors, one of which is null, with a special emphasis on black hole solutions. Besides extreme BTZ black holes and, for a special value of the coupling constant, massless null warped black holes, we also obtain for a critical coupling a family of massive "log" black holes. These are asymptotic to the extreme BTZ black holes in the sense of log gravity.
We derive the equations corresponding to twisting type-N vacuum gravitational fields with one Killing vector and one homothetic Killing vector by using the same approach as that developed by one of us in order to treat the case with two non-commuting Killing vectors. We study the case when the homothetic parameter $ϕ$ takes the value -1, which is shown to admit a reduction to a third-order real ordinary differential equation for this problem, similar to that previously obtained by one of us when two Killing vectors are present.
We show that two stationary, asymptotically flat vacuum black holes in 5 dimensions with two commuting axial symmetries are identical if and only if their masses, angular momenta, and their ``rod structures'' coincide. We also show that the horizon must be topologically either a 3-sphere, a ring, or a Lens-space. Our argument is a generalization of constructions of Morisawa and Ida (based in turn on key work of Maison) who considered the spherical case, combined with basic arguments concerning the nature of the factor manifold of symmetry orbits.
It is well known that any 4-dimensional hyperkahler metric with two commuting Killing fields may be obtained explicitly, via the Gibbons-Hawking Ansatz, from a harmonic function invariant under a Killing field on R^3. In this paper, we find all selfdual Einstein metrics of nonzero scalar curvature with two commuting Killing fields. They are given explicitly in terms of a local eigenfunction of the Laplacian on the hyperbolic plane. We discuss the relation of this construction to a class of selfdual spaces found by Joyce, and some Einstein-Weyl spaces found by Ward, and then show that certain `…
The equation of motion for comohogeneity-one Nambu-Goto strings in flat space $\mathbb{R}^{n,1}$ has been investigated. We first classify possible forms of the Killing vector fields in $\mathbb{R}^{n,1}$ after appropriate action of the Poincaré group. Then, all possible forms of the Hamiltonian for the cohomogeneity-one Nambu-Goto strings are determined. It has been shown that the system always has the maximum number of functionally independent, pair-wise commuting conserved quantities, i.e. it is completely integrable. We have also determined all the possible coordinate forms of the Killing v…
The Hamiltonian structure of spacetimes with two commuting Killing vector fields is analyzed for the purpose of addressing the various problems of time that arise in canonical gravity. Two specific models are considered: (i) cylindrically symmetric spacetimes, and (ii) toroidally symmetric spacetimes, which respectively involve open and closed universe boundary conditions. For each model canonical variables which can be used to identify points of space and instants of time, {\it i.e.}, internally defined spacetime coordinates, are identified. To do this it is necessary to extend the usual ADM …
We discuss the reduction of gravitating Chern-Simons electrodynamics with two commuting Killing vectors to a dynamical problem with four degrees of freedom, and present regular particle-like solutions.
We discuss various properties of rotating Killing horizons in generic $F(R)$ theories of gravity in dimension four for spacetimes endowed with two commuting Killing vector fields. Assuming there is no curvature singularity anywhere on or outside the horizon, we construct a suitable $(3+1)$-foliation. We show that similar to Einstein's gravity, we must have $T_{ab}k^ak^b=0$ on the Killing horizon, where $k^a$ is a null geodesic tangent to the horizon. For axisymmetric spacetimes, the effective gravitational coupling $\sim\,F'^{-1}(R)$ should usually depend upon the polar coordinate and hence ne…
Based on the CR formalism of algebraically special spacetimes by Hill, Lewandowski and Nurowski, we derive a nonlinear system of two real ODEs, of which the general solution determines a twisting type II (or more special) vacuum spacetime with two Killing vectors (commuting or not) and at most seven real parameters in addition to the cosmological constant Lambda. To demonstrate a broad range of interesting spacetimes that these ODEs can capture, special solutions of various Petrov types are presented and described as they appear in this approach. They include Kerr-NUT, Kerr and Debney/Demiańsk…
We show how one can systematically construct vacuum solutions to Einstein field equations with $D-2$ commuting Killing vectors in $D>4$ dimensions. The construction uses Einstein-scalar field seed solutions in 4 dimensions and is performed both for the case when all the Killing directions are spacelike, as well as when one of the Killing vectors is timelike. The later case corresponds to generalizations of stationary axially symmetric solutions to higher dimensions. Some examples representing generalizations of known higher dimensional stationary solutions are discussed in terms of their rod s…
Hidden symmetry transformations of D-dimensional vacuum metrics with D-3 commuting Killing vectors are studied. We solve directly the Einstein equations in the Maison formulation under additional assumptions. We relate the 4-dimensional Reissner-Nordström solution to a particular case of the 5-dimensional Gross-Perry metric.
A method of solving perfect fluid Einstein equations with two commuting spacelike Killing vectors is presented. Given a spacelike 2-dimensional surface in the 3-dimensional nonphysical Minkowski space the field equations reduce to a single nonlinear differential equation. An example is discussed.
We construct compact initial data of constant mean curvature $\widetilde{K}$ for Einstein's 4d vacuum equations with $\widehatΛ = Λ- (\widetilde{K}^2/3)$ positive, where $Λ$ is the cosmological constant, via the conformal method. To construct a transverse, trace-free (TT) momentum tensor explicitly we first observe that, if the seed manifold has two orthogonal Killing vectors, their symmetrized tensor product is a natural TT candidate. Without the orthogonality requirement, but on locally conformally flat seed manifolds there is a generalized construction for the momentum which also involves t…
Starting with a subclass of the four-dimensional spaces possessing two commuting Killing vectors and a non-trivial Killing tensor, we fully integrate Einstein's vacuum equation with a cosmological constant. Although most of the solutions happen to be already known, we have found a solution that, as far as we could search for, has not been attained before. We also characterize the geometric properties of this new solution, it is a Kundt spacetime of Petrov type II possessing a null Killing vector field and an isometry algebra that is three-dimensional and abelian. In particular, such solution b…
Static, axisymmetric solutions form a large class of important black holes in classical GR. In four dimensions, the existence of their most general metric ansatz relies on the fact that two-dimensional subspaces of the tangent space at each point spanned by vectors orthogonal to the time-translation and rotation Killing fields are integrable. This was first proved by Wald via an application of Frobenius theorem. In this note, we furnish an elementary proof for this theorem by Wald in arbitrary dimensions which yields the metric ansatz for the most general solution of the D-dimensional vacuum E…
We consider 5D Einstein-Maxwell-dilaton (EMd) gravity in spacetimes with three commuting Killing vectors: one timelike and two spacelike Killing vectors, one of which is hypersurface-orthogonal. Assuming a special ansatz for the Maxwell field we show that the 2-dimensional reduced EMd equations are completely integrable. We also develop a solution generating method for explicit construction of exact EMd solutions from known exact solutions of 5D vacuum Einstein equations with considered symmetries. We derive explicitly the rotating dipole black ring solutions as a particular application of the…
The Einstein equations for spacetimes with two commuting spacelike Killing field symmetries are studied from a Hamiltonian point of view. The complexified Ashtekar canonical variables are used, and the symmetry reduction is performed directly in the Hamiltonian theory. The reduced system corresponds to the field equations of the SL(2,R) chiral model with additional constraints. On the classical phase space, a method of obtaining an infinite number of constants of the motion, or observables, is given. The procedure involves writing the Hamiltonian evolution equations as a single `zero curvature…
We demonstrate separability of the Dirac equation in weakly charged rotating black hole spacetimes in all dimensions. The electromagnetic field of the black hole is described by a test field approximation, with vector potential proportional to the primary Killing vector field. It is shown that the demonstrated separability can be intrinsically characterized by the existence of a complete set of mutually commuting first order symmetry operators generated from the principal Killing-Yano tensor. The presented results generalize the results on integrability of charged particle motion and separabil…
The analysis of vacuum general relativity by R. Beig and N. O Murchadha (Ann. Phys. vol 174, 463 (1987)) is extended in numerous ways. The weakest possible power-type fall-off conditions for the energy-momentum tensor, the metric, the extrinsic curvature, the lapse and the shift are determined, which, together with the parity conditions, are preserved by the energy-momentum conservation and the evolution equations. The algebra of the asymptotic Killing vectors, defined with respect to a foliation of the spacetime, is shown to be the Lorentz Lie algebra for slow fall-off of the metric, but it i…
Symplectic mechanics of relativistic spinning compact bodies. I. linear-in-spin integrability under Killing-Yano symmetry
Université de Caen Normandie, ENSICAEN, CNRS/IN2P3, LPC Caen UMR6534（法国卡昂诺曼底大学、ENSICAEN、CNRS/IN2P3、LPC Caen UMR6534）；Department of Physics, National University of Singapore（新加坡国立大学物理系）
We consider 5D Einstein-Maxwell (EM) gravity in spacetimes with three commuting Killing vectors: one timelike and two spacelike Killing vectors one of them being hypersurface-orthogonal. Assuming a special ansatz for the Maxwell field we show that the 2-dimensional reduced EM equations are completely integrable by deriving a Lax-pair presentation. We also develop a solution generating method for explicit construction of exact EM solutions with considered symmetries. We also derive explicitly a new rotating six parametric 5D EM solution which includes the dipole black ring solution as a particu…
Gowdy spacetimes are generalized to admit two commuting spatial "local" Killing vectors, and some new varieties of them are presented, which are all closely related to Thurston's geometries. Explicit spatial compactifications, as well as the boundary conditions for the metrics are given in a systematic way. A short comment on an implication to their dynamics toward the initial singularity is made.
In 4 spacetime dimensions there is a well known proof that for any asymptotically flat, stationary, and axisymmetric vacuum solution of Einstein's equation there exists a "$t$-$ϕ$" reflection isometry that reverses the direction of the timelike Killing vector field and the direction of the axial Killing vector field. However, this proof does not generalize to higher spacetime dimensions. Here we consider asymptotically flat, stationary, and axisymmetric (i.e., having one or more commuting rotational isometries) black hole spacetimes in vacuum general relativity in $d \geq 4$ spacetime dimensio…
We determine the most general three-dimensional vacuum spacetime with a negative cosmological constant containing a non-singular Killing horizon. We show that the general solution with a spatially compact horizon possesses a second commuting Killing field and deduce that it must be related to the BTZ black hole (or its near-horizon geometry) by a diffeomorphism. We show there is a general class of asymptotically AdS$_3$ extreme black holes with arbitrary charges with respect to one of the asymptotic-symmetry Virasoro algebras and vanishing charges with respect to the other. We interpret these …
Stationary and axisymmetric perfect-fluid metrics are studied under the assumption of the existence of a conformal Killing vector field and in the general case of differential rotation. The possible Lie algebras for the conformal group and corresponding canonical line-elements are explicitly given. It turns out that only four different cases appear, the abelian and other three called I, II and III. We explicitly find all the solutions in the abelian and I cases. For the abelian case the general solution depends on an arbitrary function of a single variable and the perfect fluid satisfies the e…
The 3+1 Hamiltonian Einstein equations, reduced by imposing two commuting spacelike Killing vector fields, may be written as the equations of the $SL(2,R)$ principal chiral model with certain `source' terms. Using this formulation, we give a procedure for generating an infinite number of non-local constants of motion for this sector of the Einstein equations. The constants of motion arise as explicit functionals on the phase space of Einstein gravity, and are labelled by sl(2,R) indices.
We find two-dimensional free-field variables for D-dimensional general relativity on spacetimes with D-2 commuting spacelike Killing vector fields and non-compact spatial sections for D>4. We show that there is a canonical transformation which maps the corresponding two-dimensional dilaton gravity theory into a two-dimensional diffeomorphism invariant theory of the free-field variables. We also show that the spacetime metric components can be expressed as asymptotic series in negative powers of the dilaton, with coefficients which can be determined in terms of the free fields.
The Einstein equation in D dimensions, if restricted to the class of space-times possessing n = D - 2 commuting hypersurface-orthogonal Killing vectors, can be equivalently written as metric-dilaton gravity in 2 dimensions with n scalar fields. For n = 2, this results reduces to the known reduction of certain 4-dimensional metrics which include gravitational waves. Here, we give such a representation which leads to a new proof of the Birkhoff theorem for plane-symmetric space--times, and which leads to an explanation, in which sense two (spin zero-) scalar fields in 2 dimensions may incorporat…
In four dimensional general relativity, the fact that a Killing vector in a vacuum spacetime serves as a vector potential for a test Maxwell field provides one with an elegant way of describing the behaviour of electromagnetic fields near a rotating Kerr black hole immersed in a uniform magnetic field. We use a similar approach to examine the case of a five dimensional rotating black hole placed in a uniform magnetic field of configuration with bi-azimuthal symmetry, that is aligned with the angular momenta of the Myers-Perry spacetime. Assuming that the black hole may also possess a small ele…
We report on a series of new solutions to five-dimensional minimal supergravity. Our method applies to space-times with two commuting Killing symmetries and consists in combining dimensional reduction on two-spaces of constant curvature with reduction on a two-torus. The first gives rise to various generalized Bertotti-Robinson solutions supported by electric and magnetic fluxes, which presumably describe the near-horizon regions of black holes and black rings (strings). The second provides generating techniques based on U-duality of the corresponding three-dimensional sigma model. We identify…
We provide Vasiliev's four-dimensional bosonic higher-spin gravities with six families of exact solutions admitting two commuting Killing vectors. Each family contains a subset of generalized Petrov Type-D solutions in which one of the two so(2) symmetries enhances to either so(3) or so(2,1). In particular, the spherically symmetric solutions are static and we expect one of them to be gauge-equivalent to the extremal Didenko-Vasiliev solution given in arXiv:0906.3898. The solutions activate all spins and can be characterized either via generalized electric and magnetic charges defined asymptot…
The four-dimensional bosonic effective action of the toroidally compactified heterotic string incorporating a dilaton, an axion and one $U(1)$ vector field is studied on curved space-time manifolds with one and two commuting Killing vectors. In the first case the theory is reduced to a three-dimensional sigma model possessing a symmetric pseudoriemannian target space isomorphic to the coset $SO(2,3)/(SO(3)\times SO(2))$. The ten-parameter group $SO(2,3)$ of target space isometries contains embedded both $S$ and $T$ classical duality symmetries of the heterotic string. With one more ignorable c…
